Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

Let $n$ be an even positive integer. We fill in a number on each cell of a rectangle table of $n$ columns and multiple rows as following: i. Each row is assigned to some positive integer $a$ and its cells are filled by $0$ or $a$ (in any order); ii. The sum of all numbers in each row is $n$. Note that we cannot add any more row to the table such that the conditions (i) and (ii) still hold. Prove that if the number of $0$’s on the table is odd then the maximum odd number on the table is a perfect square.

Let $ABCD$ be a convex quadrilateral inscibed in circle $(O)$ such that $DB = DA + DC$. The point $P$ lies on the ray $AC$ such that $AP = BC$. The point $E$ is on $(O)$ such that $BE \perp AD$. Prove that $DP$ is parallel to the angle bisector of $\angle BEC$.

Let $ABC$ be an acute, non isosceles triangle with $M, N, P$ are midpoints of $BC, CA, AB$, respectively. Denote $d_1$ as the line passes through $M$ and perpendicular to the angle bisector of $\angle BAC$, similarly define for $d_2, d_3$. Suppose that $d_2 \cap d_3 = D$, $d_3 \cap d_1 = E,$ $d_1 \cap d_2 = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of triangle $DEF$ is the midpoint of segment $IH$.

a) For integer $n \ge 3$, suppose that $0 < a_1 < a_2 < ...< a_n$ is a arithmetic sequence and $0 < b_1 < b_2 < ... < b_n$ is a geometric sequence with $a_1 = b_1, a_n = b_n$. Prove that a_k > b_k for all $k = 2,3,..., n -1$. b) Prove that for every positive integer $n \ge 3$, there exist an integer arithmetic sequence $(a_n)$ and an integer geometric sequence $(b_n)$ such that $0 < b_1 < a_1 < b_2 < a_2 < ... < b_n < a_n$.

Two sets of positive integers $A, B$ are called connected if they are not empty and for all $a \in A, b \in B$, number $ab + 1$ is a perfect square. i) Given $A =\{1, 2,3, 4\}$. Prove that there does not exist any set $B$ such that $A, B$ are connected. ii) Suppose that $A, B$ are connected with $|A|,|B| \ge 2$. For any $a_1 > a_2 \in A$ and $b_1 > b_2 \in B$, prove that $a_1b_1 > 13a_2b_2$.

Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

A non-empty subset of $\{1,2, ..., n\}$ is called arabic if arithmetic mean of its elements is an integer. Show that the number of arabic subsets of $\{1,2, ..., n\}$ has the same parity as $n$.

Consider the function $f (x) = (x - F_1)(x - F_2) ...(x -F_{3030})$ with $(F_n)$ is the Fibonacci sequence, which defined as $F_1 = 1, F_2 = 2$, $F_{n+2 }=F_{n+1} + F_n$, $n \ge 1$. Suppose that on the range $(F_1, F_{3030})$, the function $|f (x)|$ takes on the maximum value at $x = x_0$. Prove that $x_0 > 2^{2018}$.

Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

Let $ABC$ be an acute-angled triangle inscribed in circle $(O)$. Let $G$ be a point on the small arc $AC$ of $(O)$ and $(K)$ be a circle passing through $A$ and $G$. Bisector of $\angle BAC$ cuts $(K)$ again at $P$. The point $E$ is chosen on $(K)$ such that $AE$ is parallel to $BC$. The line $PK$ meets the perpendicular bisector of $BC$ at $F$. Prove that $\angle EGF = 90^o$.

Find all positive integers $k$ such that there exists some permutation of $(1, 2,...,1000)$ namely $(a_1, a_2,..., a_{1000}) $ and satisfy $|a_i - i| = k$ for all $i = 1,1000$.